Simplifying the expression:

(B^4 + 2B^2) - B^2(B - 1)(B + 1) + 2B(3 - 2B^2)

= B^4 + 2B^2 - B^2(B^2 - 1) + 2B(3 - 2B^2)

= B^4 + 2B^2 - B^4 + B^2 + 2B(3) - 2B(2B^2)

= 2B^2 + B^2 + 6B - 4B^3

= 3B^2 + 6B - 4B^3